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area among 2 segments and an arc 2019 BMT Individual 17

Source:

January 5, 2022

number theorygreatest common divisorgeometry

Problem Statement

Let

C

be a circle of radius

1

and

O

its center. Let

\overline{AB}

be a chord of the circle and

D

a point on

\overline{AB}

such that

OD =\frac{\sqrt2}{2}

such that

D

is closer to

A

than it is to

B

, and if the perpendicular line at

D

with respect to

\overline{AB}

intersects the circle at

E

and

F

,

AD = DE

. The area of the region of the circle enclosed by

\overline{AD}

,

\overline{DE}

, and the minor arc

AE

may be expressed as

\frac{a + b\sqrt{c} + d\pi}{e}

where

a, b, c, d, e

are integers, gcd

(a, b, d, e) = 1

, and

c

is squarefree. Find

a + b + c + d + e

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